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- Ruby has the following utility function: U(X, Y) = X^3/4 , Y^1/4, where X is her consumption of food items, with a price of $10, and Y is her consumption of clothing items, with a price of $30. She plans to spend $360 on food and clothing over the next week. Using the Lagrange multiplier technique, determine the number of food and clothing items that will maximize Ruby's utility.1. Use the Method of Lagrange to solve this problem. To do so, construct the La- grangean function for this problem. Use λ1 as the Lagrange multiplier attached to the period 1 budget constraint and λ2 as the Lagrange multiplier attached to the period 2 budget constraint.Consider U(q1,q2) = q1 + v(q2) where v' > 0 and v'' < 0. This utility function is called a quasi-linear utility function. Assume q1 is a numeraire. Find the demand function for q2. *What does v mean in this question? Also, could you solve this problem without using Lagrange multipliers? Thank you.
- Mr Banda faces the following utility function from consuming nshima [X] and Rice [Y] U = 3XY 1 3 Mr Banda’s budget allocation for nshima and rice is K240. The price of nshima is K60 per Kg while that of Rice is K20 per Kg. A. How much nshima and rice should Mr. Banda consume to maximize his utility and What is the total utility at the optimum? b. What is the Lagrange multiplier? c. What will be the increase in utility when the budget allocation is increased to K241?A consumer consumes two agricultural products: Red Meat, and Tomatoes according to the following utility function: U = RT That is, the total utility is the multiplication of the quantity consumed of the two products. Given that consumer's income is 210, price of R is 10, and the price of Tis 2, a)Write down the budget constraint (budget line equation) for this consumer. b)Determine the quantities that the consumer should consume of each of the two products c)Calculate the value of the lagrangian multiplier and derive the demand function for Red Meat and for tomatoes.1.2) Suppose that a consumer’s utility function is U=10 lnx+20 lny. A) Find the marginal utility of x, MU_x, and the marginal utility of y, MU_y.B) Suppose that the consumer has R3600 to spend on x and y, while p_x=200 and p_y=400. C) Use the Lagrange multiplier (LM) method to find the levels of x and y that will maximise the consumer’s utility, subject to her budget constraint. D) Find and interpret the value of the Lagrange multiplier. Show that MRCS=p_x/p_y at the utility maximising levels of x and y.
- Q1) Sales are a function of advertising in newspapers and magazines (X, Y). S = XY2 The price of advertising in newspapers and magazines is Rs.5 and Rs.10 respectively. The total budget for advertising is Rs.105. For maximizing sales, find out the best combination of advertisements in newspapers and magazines by using the Lagrangian multiplier. (Hint: Make equation of the budget line with the help of the above information).You are given the following utility function: ? = ?? The budget is K100 and the price of X is K2 while the price of Y is K5. a) Derive the demand for X and Y by the Lagrange multiplier method. b) What will be the demand when all the prices are doubled and the income is doubled? c) What is the utility when the budget is increased by K1?Sales are a function of advertising in newspapers and magazines (X, Y). 05 S = XY2 Price of advertising in newspapers and magazines are Rs.5 and Rs.10 respectively. The total budget for advertising is Rs.105. For maximizing the sales, find out the best combination of advertisement in newspapers and magazines by using lagrangian multiplier. (Hint: Make equation of budget line with the help of above information).
- Zeynep lives on an island where she produces two goods, apples (x) and bananas (y), according to the production possibility frontier 200 = x² + y², and she consumes all the goods herself. Her utility function is U(x,y) = xy³. Find her utility maximizing x and y as well as the value of ? (Lagrange multiplier).The utility derived by a consumer from the consumption of two commodities is given by the function U (A, B) = 0.5In (A) + 0.5 In (B) where A are the number of units of the first commodity consumed and B are the number of units of the second commodity consumed each month. A unit of the first commodity costs $8 and a unit of the second commodity costs $ 4 using the Lagrange multiplier method determine the optimal quantity of each of the commodities consumed each month given that consumer has $32 to spend on both commodities each monthMichael does not like to mix peanut butter and jelly in the same sandwich. However, he will consume them separately; for him, a sandwich with 1 spoon of peanut butter is exactly the same as a sandwich with 2 spoons of jelly. Michael has an income of m = 50, and the prices per spoonful of peanut butter and jelly are pPB=5 and pJ=11. Please write down Michael’s utility function over peanut butter (PB) and jelly (J). 2. Please determine Michael’s Marshallian demands PB*m and J*m3. Please determine Michael’s new Marshallian demands PB*m and J*m, when the price of peanut butter falls to pPB = 1.4. What are the (Hicks) SE and IE? Draw a diagram to show your analysis, with peanut butter on thehorizontal axis, and jelly on the vertical axis. 5. Recall that there are two different types of substitution effects. For example, in Q2 we have used theHicks SE. Does your answer to the last part change if we use the Slutsky SE? Justify. 6. Nam likes his peanut butter and jelly sandwiches with exactly…
